A solution to a differential equation at an interval α < t < b is any function y (t) that satisfies the differential equation in question on the interval α < t < b.  It is significant to note that solutions are frequently accompanied through intervals and these intervals can impart several of significant information regarding the solution.  Consider the subsequent illustration.

Illustration 1:  Show that y(x) = x^-3/2 is a solution to 4x² y′′ + 12xy′ + 3 y = 0 for x > 0.

Solution: We'll require the first and second derivative to do this.

y'(x) = (-3/2) x^-5/2                     y''(x) = (15/4) x^-7/2

Plug these in addition to the function in the differential equation.

4x² ((15/4) x^-7/2) + 12x((-3/2) x^-5/2) + 3(x^-3/2) = 0